Of course to be able to execute a package you must have the privilege to execute the package.
Isn't this obvious?

Why don't you directly ask your DBA?
Does he/she scare you?
Maybe you don't know how to speak to a DBA.
Never say him/her: "give me privilege to execute this package". This tends to turn the DBA against this.
Tell him/her exactly what you posted: "I have to online redefine a table, I tried to execute dbms_redefinition package, I got this error, maybe I don't have the privilege to do it, can I have it". There you will have a pretty good chance to get it.

Apart from execute grant on the package, we need some other access to perform ONLINE REDEFINITION
[code]
A user needs execute privilege on dbms_redefinition, alter any table privilege, create any table privilege, drop any table privilege, lock any table privilege and select any table privilege to effectively use this package.
[code]
I am able to compelte the task now after these grants.
Source is http://www.mydatabasesupport.com/dbms-redefinition-package.html